1. Field
The embodiments of the invention relate to computing inventory control levels of one or more items in a population of items.
2. Description of the Related Art
An inventory management system for a single site typically manages each item using two control levels: an item's reorder point (ROP), which determines when to order the item, and a requisitioning objective (RO), which determines how much of the item to order. An order is placed when assets on-hand plus on-order decrease to or below the ROP, and the difference between the RO and the current assets is the quantity ordered. If this comparison is made continuously, the inventory control system is known as a continuous review (s, S) policy, where s is the ROP and S is the RO. It is well known that when managing items at a single site, with a positive (or positive plus linear) cost to replenish stock, that an (s, S) policy is optimal. See Scarf, H. “The Optimality of (s, S) Policies in the Dynamic Inventory Problem,” Mathematical Methods in the Social Sciences, Chapter 13, Stanford University Press, 1960; Axsäter, S. Inventory Control, 2nd edition, Springer, New York, 2006, p 140; and Zipkin, P. Foundations of Inventory Management, McGraw Hill, Boston, 2000, pp. 395-401. In the case where there is no cost to replenish, an S or (S−1, S) policy is optimal. A case where there is only a positive cost to replenish stock can be considered.
Although the (s, S) policy is optimal, it is computationally intensive to find the optimal values of s and S, and simpler methods are often used. In practice, the RO is often the ROP plus a nominal order quantity, Q, which is often a Wilson Lot-size formula (EOQ) (“economic order quantity”)—this is known as (s, Q) or (R, Q) policy—see Axsäter, pp. 140-142. Thus Q is the quantity ordered if assets drop exactly to the ROP. The ROP is an estimate of lead-time demand (LTD) plus a safety level that protects against variability in LTD. An (s, Q) policy works well in the case of unit or constant demand sizes, where the inventory position decreases exactly to the ROP and triggers a replenishment order. In the case of variable customer demand sizes, sometimes known as “lumpy demand”, a demand may be large enough to reduce the inventory position from above the ROP to below the ROP by more than Q units. In this case an (s, nQ) policy is sometimes used, where the replenishment order size is the smallest multiple of Q that restores the inventory position to a value above s—see Zipkin, p. 227.
For each type of policy, it is then necessary to find optimal, or at least reasonable values of s and S, or s and Q for each item. Optimality is defined with respect to metrics for customer service, inventory investment, and replenishment cost. Customer service is typically measured in terms of expected (average) backorders, customer wait time, or service level. Inventory investment is typically measured as the average value of inventory on hand, or average value of the inventory position. Replenishment cost is typically measured in terms of the number of replenishment actions per unit time, multiplied by a cost to place a replenishment request. Typically the objective function for the optimization is the sum of terms for (1) expected costs to order replenishment per unit time, (2) expected inventory value, and (3) expected backorders, multiplied by a backorder penalty factor—see Zipkin, p. 213.
Expected costs of inventory and of backorders are computed from the probability distribution of on hand inventory, where negative values denote backordered units. The probability distribution for on-hand inventory is computed from the probability distribution of the inventory position and the probability distribution of leadtime demand. For unit or constant demand sizes, the inventory position is uniformly distributed. For variable demand sizes the distribution of the inventory position is uniform for the (s, nQ) policy with certain restrictions on the demand size distribution—see Zipkin, p. 230, but for the (s, S) policy the inventory position probabilities are not uniform and are computed via renewal equations involving the probabilities for demand sizes—see Sahin, I. “On the Stationary Analysis of Continuous Review (s, S) Inventory Systems,” Operations Research, Vol. 27, 1979, pp 717-729; Zheng, Y. and Federgruen (Z-F), A. “Finding Optimal (s, S) Policies is About as Simple as Evaluating a Single Policy”, Operations Research, Vol. 39, 1991, pp. 654-665; and Zheng, Y. and Federgruen, “Errata: Finding Optimal (s, S) Policies is about as Simple as Evaluating a Single Policy,” Operations Research, Vol. 40, 1992, p. 192. For (s, Q), (s, nQ) and (s, S) policies, a leadtime demand distribution is typically assumed (e.g., Poisson, negative binomial, Normal, Gamma, Weibull), and its mean and variance are estimated—see Axsäter, pp. 140-145. An alternative is to estimate the leadtime demand probabilities via bootstrapping—see Fricker, R., and Goodhart, C., “Applying a Bootstrap Approach for Setting Reorder Points in Military Supply Systems,” Naval Research Logistics, vol. 47, no. 6 (2000): 459-478.
In summary, in the case of (s, Q) or (s, nQ) policies, expected replenishment costs per unit time are typically computed using the theoretical probability distribution for demand sizes and the assumed Q—see Zipkin, p. 229. For (s, S) policies, expected replenishment costs are computed from the renewal density, derived from the theoretical probability distribution of demand sizes, as well as from s and S—see Sahin and Zheng. In addition, expected cost of holding inventory and expected cost of backorders could be computed from the theoretical probability distribution of on hand inventory, where negative values for on hand inventory denote backordered units. Therefore, a related method for determining optimal inventory control levels (s, S) assumes theoretical probabilities, including for leadtime demands, and assumes an average demand size. In other words, the related method assumes probability distributions, which requires estimating all of the parameters (number of demands in a leadtime, time of (mean) demand, size of demand (requisition size)) to yield assumed probability distributions. However, this introduces errors in both the estimating of parameters and the assumed probability distributions.
Optimization methods—see Zipkin, pp. 223-224 and Zheng—are then applied to the cost function to solve for the values of s and Q, or values of s and S for each item that minimize overall expected costs.
The state of the system typically changes over time as demand patterns for individual items experience fluctuations or trends. This is typically treated by periodically updating the mean and variance estimates for the assumed probability distribution of leadtime demands, re-computing the expected costs, and re-running the optimization.
A limitation of the (s, nQ) policy is that it is not the optimal form of policy, so for populations of items with certain characteristics, the use of such a policy will result in excessive inventory, inferior customer service, and excessive replenishment workload.
A limitation of the approach described earlier for optimizing inventory levels under an (s, S) policy is that for a population of items in which the demand sizes vary over a wide range, the computation of items' inventory position probabilities may be impractical, based on computational resources required.
A limitation of the (s, S) inventory levels computation described earlier is that, for items in certain populations, computationally tractable theoretical distributions do not fit actual distributions of leadtime demand well. This leads to an inaccurate assessment for the costs of backorders and inventory, and therefore results in inventory levels that produce excessive inventory, inferior customer service, and excessive replenishment workload.
A limitation of the (s, S) inventory levels computation described earlier is that, for items in certain populations, parameter estimates for theoretical distributions are subject to errors as large as 100 to 200 percent.
A limitation of the bootstrapping approach for developing a probability distribution of leadtime demand is that for large item populations, it is impractical, based on computational resources required.
A limitation of the (s, S) inventory levels computation described earlier is that, for large populations of items, the computation of the cost function is impractical, based on computational resources required.
A limitation of the (s, S) inventory levels computation described earlier is that the cost function penalizes only expected units backordered, not expected requisitions backordered. This results in a significant imbalance in customer service and investment across items with small and large demand sizes, leading to excessive inventory for items with large demand sizes and inferior customer service for items with small demand sizes.
A limitation of the (s, S) inventory levels computation described earlier is that the optimization algorithm may invest heavily in inventory for a small number of items, resulting in inferior customer service for the majority of items in the population.